CN102710288B

CN102710288B - Orthogonal PSWF (Prolate Spheroidal Wave Function) pulse design method on basis of cross-correlation matrix diagonalization

Info

Publication number: CN102710288B
Application number: CN201210210645.9A
Authority: CN
Inventors: 赵志勇; 王红星; 张晨亮; 刘锡国; 钟佩琳; 康家方; 陈昭男
Original assignee: Individual
Current assignee: School Of Aeronautical Combat Service Naval Aeronautical University Of People's Liberation Army
Priority date: 2012-06-15
Filing date: 2012-06-15
Publication date: 2015-06-17
Anticipated expiration: 2032-06-15
Also published as: CN102710288A

Abstract

The invention discloses an orthogonal PSWF (Prolate Spheroidal Wave Function) pulse design method on the basis of the cross-correlation matrix diagonalization, which is the orthogonal PSWF pulse design method. The orthogonal PSWF pulse design method comprises the following steps of: by calculating a cross-correlation matrix of a PSWF pulse which participates in the orthogonalization design, carrying out diagonalization transformation on the cross-correlation matrix to obtain an orthogonal matrix; and then carrying out matrix multiplication operation by a transposed matrix of the orthogonal matrix and the PSWF pulse so as to implement the orthogonalization of the PSWF pulse. According to the method, the calculating time complexity and the storage space of the orthogonal PSWF pulse design are reduced; and the method is not only suitable for the orthogonal design of the base band PSWF pulse, but also suitable for the orthogonal design of the band pass PSWF pulse.

Description

Orthogonal PSWF pulse design method based on cross-correlation matrix diagonalization

Technical Field

The invention relates to a waveform design method in radio communication, in particular to a time domain orthogonal pulse design method.

Background

The elliptic spherical Wave function set (PSWF) has the excellent characteristics of time-frequency domain energy aggregation optimization, time domain biorthogonal, completeness, approximate time limit band limit, spectrum controllability and the like, is widely concerned by academia from the beginning and shows good application prospect. Slepian and Pollak et al first proposed and studied the PSWF function set in the Bell laboratory study report of 1961, and subsequently published a series of related study reports over the next 20 years.

The following two types of PSWF definitions are mainly used.

(1) Differential equation definitional equation

The differential equation for PSWF is defined as follows:

wherein psi_n(c, t) is n-order PSWF, χ_nThe characteristic value corresponding to the n-order PSWF, c is the time-bandwidth product of the PSWF, and T is the duration width of the PSWF.

(2) Integral equation definitional equation

<math> <mrow> <msubsup> <mo>&Integral;</mo> <mrow> <mo>-</mo> <mi>T</mi> <mo>/</mo> <mn>2</mn> </mrow> <mrow> <mi>T</mi> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mi>h</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mi>dτ</mi> <mo>=</mo> <msub> <mi>λ</mi> <mi>n</mi> </msub> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </math>

Wherein h (t) is the kernel function of PSWF, λ_nIs an n-order PSWF psi_nAn energy concentration factor of (c, t).

If the kernel function h (t) has an ideal low-pass characteristic, that is, h (t) is sin Ω t/π t, Ω is an angular frequency, the frequency domain characteristic is:

the integral equation for PSWF is defined as:

<math> <mrow> <msubsup> <mo>&Integral;</mo> <mrow> <mo>-</mo> <mi>T</mi> <mo>/</mo> <mn>2</mn> </mrow> <mrow> <mi>T</mi> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mfrac> <mrow> <mi>sin</mi> <mi>Ω</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>π</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mi>dτ</mi> <mo>=</mo> <msub> <mi>λ</mi> <mi>n</mi> </msub> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> </math>

in this case, since the frequency domain energy accumulation region of the PSWF is [ - Ω, + Ω ], the PSWF defined by equation (4) is generally referred to as a baseband PSWF.

② corresponding to this, if h (t) has ideal band-pass characteristics, i.e. kernel function is:

h(t)＝sinΩ_Ht/πt-sinΩ_Lthe frequency domain characteristic of t/pi t (5) is as follows:

the integral equation for PSWF is defined as:

<math> <mrow> <msubsup> <mo>&Integral;</mo> <mrow> <mo>-</mo> <mi>T</mi> <mo>/</mo> <mn>2</mn> </mrow> <mrow> <mi>T</mi> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>[</mo> <mfrac> <mrow> <mi>sin</mi> <msub> <mi>Ω</mi> <mi>H</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>π</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <mi>sin</mi> <msub> <mi>Ω</mi> <mi>L</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>π</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>]</mo> <mi>dτ</mi> <mo>=</mo> <msub> <mi>λ</mi> <mi>n</mi> </msub> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow> </math>

in this case, the PSWF has a frequency domain energy accumulation interval of [ omega ]_L,Ω_H]Therefore, the PSWF defined by the formula (7) is generally referred to as a bandpass PSWF.

Currently, PSWF has been widely used in many fields. The function is used for wavelet signal analysis, and can achieve higher time resolution than a sinc function; the method is used for processing digital images and digital signals, can effectively solve the contradiction between time resolution and space resolution, is used for analyzing and modeling an optical system, and has better universality; the method is used for wireless communication channel research, and can more accurately model a high-speed mobile communication channel and a time-frequency selective fading channel; the method is used for communication signal design, can realize flexible spectrum control and has higher spectrum efficiency and power efficiency.

However, PSWF does not have an analytical solution, and a numerical solution method is usually used to obtain an approximate solution. Among the approximate solutions for PSWF, there are mainly Parr numerical solutions (see Parr B, Cho B, Wallace K.A novel-wireless basic design equation [ J ]. IEEE Communication Letters,2003,7(5): 219-), reconstruction solutions (see Khare K, George N.sampling the approach to template spherical wave functions [ J ]. Journal of Physics,2003,36(39): 10011-), and the like. These solving methods all involve a process of solving the matrix eigenvector, and thus have a problem of high implementation complexity. In the Legendre polynomial approximation method-based literature (see the literature: Wangdong, Chengzhao man, Zhaoyong, etc., Legendre polynomial-based ellipsoidal wave pulse design method [ J ] in the science of electric waves, 2012,27(1):191-197.), Wangdong and the like, a low-complexity approximate solution method is provided for the baseband PSWF. The method calculates the coefficient of the normalized Legendre polynomial according to the PSWF parameters, and obtains the approximate numerical solution of the PSWF through the weighted summation of the normalized Legendre polynomial.

In order to effectively improve the power efficiency and the spectral efficiency of a communication system, the invention patent (application number 200810237849.5) discloses a time domain orthogonal and channel overlapping orthogonal pulse design method, and the designed orthogonal pulse can enable the communication system to have higher frequency band utilization rate and better energy aggregation property. In the patent, the whole pulse design frequency band is divided into a plurality of mutually overlapped sub-frequency bands, each-order PSWF pulse is solved on each sub-frequency band by using a Parr numerical solution, and finally the orthogonal PSWF pulse is obtained by a Schmidt orthogonalization method. The method has higher computation time complexity and storage space, thereby increasing the requirement on the hardware level.

Disclosure of Invention

In order to overcome the defects of the prior art, the invention discloses an orthogonal PSWF pulse design method based on cross-correlation matrix diagonalization. According to the method, the orthogonal PSWF pulse design is realized by carrying out diagonalization transformation on the cross-correlation matrix of the PSWF pulse participating in the orthogonalization design.

In the invention, for the orthogonalization design of the baseband PSWF, on the basis of a Legendre polynomial-based elliptical spherical wave pulse design method, an approximate numerical solution of the PSWF is solved by adopting normalized Legendre polynomial (Legendre) polynomial approximation, and the numerical solution and the orthogonalization process of the PSWF are optimized and integrated so as to reduce the implementation complexity. For the orthogonalization design of the band-pass PSWF, the invention adopts a diagonalization method based on a cross-correlation matrix to replace a Schmidt orthogonalization method. The method can effectively reduce the complexity of the calculation time and the complexity of the storage space.

The technical measures of the present invention are explained in detail below from two aspects of baseband PSWF quadrature pulse design and bandpass PSWF quadrature pulse design, respectively, to achieve the object of the present invention.

(1) Baseband PSWF quadrature pulse design

According to the method, the PSWF pulse solving algorithm based on the normalized Legendre polynomial approximation and the orthogonalization method based on the cross-correlation matrix diagonalization are analyzed, the approximate numerical solving and orthogonalization processes of the PSWF are optimized and combined, and the direct corresponding relation between the orthogonal PSWF pulse and the Legendre polynomial is established, so that the orthogonal PSWF pulse approximated by the normalized Legendre polynomial can be directly obtained by calculating the weighted sum matrix of the orthogonal PSWF pulse, and the implementation complexity is reduced.

PSWF pulse solving algorithm based on normalized Legendre polynomial approximation

As known from the literature, "method for designing an ellipsoidal spherical wave pulse based on Legendre polynomial", based on normalized Legendre polynomial expansion, the jth order PSWF can be expressed as:

<math> <mrow> <msub> <mi>ψ</mi> <mi>j</mi> </msub> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mi>Σ</mi> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mo>∞</mo> </munderover> <msubsup> <mi>β</mi> <mi>k</mi> <mi>j</mi> </msubsup> <mo>·</mo> <msub> <mover> <mi>P</mi> <mo>&OverBar;</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>0,1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>M</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow> </math>

wherein,for the k-th normalized Legendre polynomial, its coefficient vector beta^jIs a feature vector of the matrix A, i.e.

Aβ^j＝χ_jβ^j(9) Wherein, the matrix A is defined as follows:

the other elements are 0. Using psi ═ psi₀(c,t),ψ₁(c,t),…,ψ_M-1(c,t)]^TDenotes a vector formed by an M-th order PSWF, B ═ β¹，β²,…,β^M]Representing a Legendre polynomial weighting coefficient matrix,representing a normalized Legendre polynomial vector, equation (8) can be expressed in the form of a matrix multiplication:

ψ＝BP (13)

orthogonalization method based on cross-correlation matrix diagonalization

For a plurality of PSWF pulses, the cross-correlation matrix of the pulses serves as a statistical index, and the correlation can be reflected most intensively. The diagonalization method based on the cross-correlation matrix starts from the cross-correlation matrix, eliminates the correlation among different pulses and realizes the orthogonalization of the pulses. A set of correlation signals is orthogonally transformed to always eliminate the correlation between the components to some extent. The cross-correlation matrix diagonalization method is an orthogonal transformation based on a cross-correlation matrix, and is optimal in view of the performance of completely eliminating the correlation of each component.

PSWF pulses are assumed to be overlapped by M frequency spectra_i(C, t), the cross-correlation matrix C of the pulse is:

wherein, c_i,jAs a function of the cross-correlation of the ith and jth pulses, i.e.The cross-correlation matrix C is a symmetric matrix, and then an orthogonal matrix X is necessary, so that X is^TAnd CX ═ Λ, wherein Λ is a diagonal matrix taking M eigenvalues of C as diagonal elements, and X is a transformation matrix of the cross-correlation matrix diagonalization method. Let X be ═ X₁,x₂,…,x_M]，x_iThe eigenvectors of the cross-correlation matrix C are

Wherein eta is_iIs the ith characteristic value of C. Let the conversion factor x_i＝[x_i1,x_i2,…,x_iM]^TCarrying out cross-correlation matrix diagonalization transformation on the M PSWF pulses to obtain a new pulse psi'_i(c, t) is

Novel pulse psi'_i(c, t) and ψ'_jThe cross-correlation function of (c, t) is

<math> <mrow> <msubsup> <mo>&Integral;</mo> <mrow> <mo>-</mo> <mi>T</mi> <mo>/</mo> <mn>2</mn> </mrow> <mrow> <mi>T</mi> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>ψ</mi> <mi>i</mi> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <msubsup> <mi>ψ</mi> <mi>j</mi> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>dt</mi> <mo>=</mo> <munderover> <mi>Σ</mi> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <munderover> <mi>Σ</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <msub> <mi>x</mi> <mi>in</mi> </msub> <msub> <mi>x</mi> <mi>jm</mi> </msub> <msubsup> <mo>&Integral;</mo> <mrow> <mo>-</mo> <mi>T</mi> <mo>/</mo> <mn>2</mn> </mrow> <mrow> <mi>T</mi> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <msub> <mi>ψ</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>dt</mi> <mo>=</mo> <munderover> <mi>Σ</mi> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <munderover> <mi>Σ</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <msub> <mi>x</mi> <mi>in</mi> </msub> <msub> <mi>x</mi> <mi>jm</mi> </msub> <msub> <mi>c</mi> <mi>mn</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow> </math>

According to X^TCX ═ Λ, available

<math> <mrow> <msubsup> <mo>&Integral;</mo> <mrow> <mo>-</mo> <mi>T</mi> <mo>/</mo> <mn>2</mn> </mrow> <mrow> <mi>T</mi> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>ψ</mi> <mi>i</mi> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <msubsup> <mi>ψ</mi> <mi>j</mi> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>dt</mi> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>η</mi> <mi>i</mi> </msub> </mtd> <mtd> <mi>i</mi> <mo>=</mo> <mi>j</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>i</mi> <mo>&NotEqual;</mo> <mi>j</mi> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow> </math>

Equation (18) demonstrates that the new pulses obtained by diagonalizing the cross-correlation matrix are orthogonal to each other. On the basis, normalization processing is carried out on the obtained orthogonal new pulse, and the PSWF pulse of the standard orthogonality is obtained.

Integration processing method for approximate numerical solution and orthogonalization process of PSWF

Is made of psi'₀(c,t),ψ′₁(c,t),…,ψ′_M-1(c,t)]^TExpressing orthogonal PSWF vectors, the PSWF pulse orthogonalization process expressed by equation (16) can be expressed as a matrix multiplication:

ψ′＝X^Tψ (19)

by substituting the PSWF solving process of equation (13) into equation (19), we can obtain:

ψ′＝X^TBP (20)

let D be X^TB, obtaining a normalized Legendre polynomial fitting form of the orthogonal PSWF pulse as follows:

ψ′＝DP (21)

referred to herein as the weighted summation matrix of the quadrature PSWF pulses, using D_iRepresenting the ith row of the matrix D, the ith quadrature PSWF pulse can be represented as

<math> <mrow> <msubsup> <mi>ψ</mi> <mi>i</mi> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mi>Σ</mi> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msub> <mi>d</mi> <mi>ik</mi> </msub> <msub> <mover> <mi>P</mi> <mo>&OverBar;</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow> </math>

According to the analytical formula (22), the method disclosed by the invention can directly establish the corresponding relation between the orthogonal PSWF pulse and the Legendre polynomial, and simplifies the design steps. Therefore, the orthogonal pulse design method disclosed by the invention reduces the implementation complexity by carrying out optimization and integration processing on the PSWF approximation value solving and orthogonalization processes, thereby improving the design efficiency of the orthogonal PSWF pulse.

(2) Orthogonal pulse design method of band-pass PSWF

The invention patent (application number 200810237849.5) discloses a time domain orthogonal, channel-overlapped orthogonal pulse design method. In the patent, a frequency spectrum width occupied by a communication channel, namely a pulse, is divided into a plurality of sub-channels which have the same bandwidth and are mutually overlapped by 50%, approximate numerical solutions of PSWF of each order are solved on each sub-channel by using a Parr numerical solution, and finally, an orthogonal PSWF pulse is obtained by a Schmidt orthogonalization method. However, as the number of pulses participating in orthogonal design increases, the orthogonality of the designed pulses is reduced by the Schmidt orthogonalization method, which degrades the interference resistance of the system when used for transmission.

The orthogonalization method based on cross-correlation matrix diagonalization disclosed by the invention replaces the Schmidt orthogonalization process in the invention patent (application number 200810237849.5) to improve the performance of designed orthogonal pulses. For the orthogonal pulse design of the band-pass PSWF, the design method disclosed by the invention comprises the following steps of: the method comprises the steps of firstly, wave channel division, secondly, parameter setting, thirdly, constructing a characteristic function and an integral equation, fourthly, solving the equation, and fifthly, orthogonalizing based on cross-correlation matrix diagonalization.

Wave channel division

Dividing the frequency spectrum width occupied by the communication channels, namely orthogonal pulses into k sub-channels with the same bandwidth and overlapped mutually in a uniform overlapping mode;

② parameter setting

Setting parameters of orthogonal pulse, frequency spectrum width B occupied by orthogonal pulse, frequency spectrum lower limit frequency f_LSum frequencyUpper limit frequency f of spectrum_HDetermining the division number k of sub-channels, a time-bandwidth product factor c of an elliptic spherical wave function and the frequency spectrum overlapping degree rho;

construction of characteristic function and integral equation

Constructing a characteristic function and an integral equation aiming at each sub-channel, wherein the constructed characteristic function is h_k(t)＝2f_k,H sinc(2f_k,Ht)-2f_k,L sinc(2f_k,Lt) wherein f_k,L、f_k,HRespectively the lower limit and the upper limit of the frequency of the kth sub-channel, and the constructed integral equation isWherein λ is_kCharacteristic function h for solving integral equation for numerical algorithm based on eigenvalue decomposition_k(t) eigenvalues of the constructed matrix,. psi_k(c, t) is lambda_kCorresponding characteristic function, T is the value range [ -T/2, T/2]A time variable of (d);

solving an equation, namely respectively solving an integral equation constructed by each sub-channel through a Parr numerical algorithm based on characteristic value decomposition, and obtaining an elliptic spherical wave function pulse of each sub-channel by taking characteristic functions corresponding to the first m maximum characteristic values;

and the orthogonalization based on the diagonalization of the cross-correlation matrix means that the cross-correlation matrix of each sub-channel PSWF pulse participating in the orthogonalization design is calculated and subjected to diagonalization transformation, so that the orthogonalization of the PSWF pulse is realized.

The orthogonalization of the fifth step based on diagonalization of cross-correlation matrix is described in detail in the baseband PSWF pulse quadrature design of this patent and will not be described here.

Compared with the prior art, the invention has the following beneficial effects:

low complexity of calculation time

Suppose that M PSWF pulses need to be orthogonalized, each PSWF pulseThe number of sampling points of the pulse is K. In the process of solving the PSWF function by utilizing Legendre polynomial approximation, the number of sampling points of each order Legendre polynomial is also K, and for each PSWF pulse, the number of required Legendre polynomials is KWhereinMeaning rounded up, c is the time-bandwidth product of PSWF, and e is the base of the natural logarithm. The construction of one orthogonal PSWF pulse requires the weighted summation of N Legendre polynomials, and therefore, NK operations (mainly multiplication) are required, and the total operation amount of the whole process is MNK.

For the design method disclosed in the patent (application No. 200810237849.5), the time complexity is mainly focused on two steps of PSWF pulse solving and Schmidt orthogonalization. In the pulse solving process based on the Parr algorithm, the operation amount comes from the process of solving the characteristic vector of the matrix. When the classical Jacobi method is adopted to solve the eigenvector, the operation amount of a K multiplied by K dimensional matrix is about K³. The Schmidt method firstly calculates a weighting coefficient a_i,jFollowed by a weighted summation process. The generation of the ith orthogonal pulse requires the calculation of i-1 weighting coefficients, and the M orthogonal pulses require the calculation of (M-1) M/2 weighting coefficients. The calculation of a weighting coefficient requires 2K +1 operations, and the total operation amount of the weighting coefficient calculation process is (M-1) (2K +1) M/2. For the weighted sum process, (i-1) K multiplications are required for the ith new pulse, and the sum process operation for all pulses is (M-1) MK/2. The operation amount of the whole Schmidt orthogonal process is (M-1) (3K +1) M/2. The temporal complexity of the patent (application No. 200810237849.5) is: c_t＝(M-1)(3K+1)M/2+K³。

When generating pulses using the Parr algorithm, the number of sampling points required for a pulse is assumed to be 4 times the sampling rate of the signal bandwidthAccording to the patent (application number 200)810237849.5), the number of pulses for orthogonal design in each sub-band is as followsBased on the above conditions, the time complexity of the two orthogonal PSWF pulse design methods was comparatively analyzed, as shown in fig. 1. As can be seen from the simulation results of fig. 1, the time complexity of the present invention is significantly smaller than that of the method of the patent (application No. 200810237849.5), and the time complexity is substantially unchanged as the time-bandwidth product increases, while the time complexity of the patent (application No. 200810237849.5) increases rapidly.

② the computing storage space is small

From the space complexity in the FPGA hardware implementation process, the main storage variables of the invention are Legendre polynomial coefficients and a fitting matrix D. The polynomial set needs to store N × K data in total, and the matrix D has M × N dimensions, so the spatial complexity S of the method₁Comprises the following steps:

S₁＝(M+K)N (23)

for the patent (application No. 200810237849.5), the PSWF calculation process constructs a feature matrix with dimension K × K, and the resulting M PSWF pulses need to be represented by M × K data. In the Schmidt quadrature procedure, two PSWF pulses need to be stored. When pulse orthogonalization is carried out, the space occupied by the characteristic matrix in the previous pulse solving process can be released, and meanwhile, the number of sampling points of each pulse is far greater than the number of pulses, namely K > M, so that compared with the pulse solving process, the space complexity in the Schmidt process can be ignored, the space complexity in the patent (application No. 200810237849.5) is mainly determined by the pulse solving process, and the expression is as follows:

S₂＝(K+M)K (24)

when the value range of c is [0,40], the space complexity required by the two pulse design methods in the FPGA hardware implementation process is shown in FIG. 2.

As can be seen from the comparison of the spatial complexity in fig. 2, the amount of data that needs to be stored is significantly reduced when the present invention is employed. The amount of spatial complexity increase of the present invention is also significantly less than that of the patent (application No. 200810237849.5) as the bandwidth product over time increases. At a time-bandwidth product of 40, the spatial complexity of the present invention is less than half that of the patent (application No. 200810237849.5). The invention has great advantages in view of space complexity.

Wide application range

The design method disclosed in the patent application No. 200810237849.5 adopts the Schmidt orthogonalization method to realize the orthogonal PSWF pulse design, and as can be seen from the characteristics of the Schmidt orthogonalization process, the requirements of the orthogonalization process are relatively strict, and the PSWF pulses participating in the orthogonalization design are strictly linearly independent. .

The correlation matrix diagonalization method disclosed by the invention only performs linear combination, namely reconstruction in the mathematical sense, on the original PSWF pulse, and does not perform fundamental change on the PSWF pulse. Meanwhile, the method has no strict requirement on the cross correlation of the pulses, and is suitable as long as the pulses of the original pulses are not in a linear relation. Therefore, the method disclosed by the invention has wide applicability.

Drawings

FIG. 1 is a graph comparing the time complexity of the two methods of the present invention and the prior art.

Fig. 2 is a comparison of the spatial complexity required for the two methods of the present invention and the prior art.

Fig. 3 is a flow chart of the baseband quadrature PSWF pulse waveform generation disclosed in the present invention.

Fig. 4 is a flow chart of the generation of the bandpass quadrature PSWF pulse waveform disclosed in the present invention.

Detailed Description

The present invention is described in further detail below with reference to the attached drawings.

(1) Baseband PSWF orthogonalization design

For the baseband PSWF orthogonalization design, the orthogonal PSWF pulse generation flow based on cross-correlation matrix diagonalization is shown in fig. 3, and the pulse generation can be performed as follows.

Sub-band division and parameter setting

Firstly, dividing a frequency spectrum width occupied by a communication channel, namely a pulse into k sub-frequency bands with the same bandwidth and overlapped with each other by 50%, wherein k is a positive integer larger than 0; then setting parameters of orthogonal PSWF pulse, the frequency spectrum width B occupied by orthogonal pulse, and the lower limit frequency f of frequency spectrum_LAnd the upper frequency f of the spectrum_HDetermining the number k of sub-channel divisions, determining the time-bandwidth product factor c of the elliptic spherical wave function, determining the frequency spectrum width B and the frequency spectrum lower limit frequency f_LAnd the upper frequency f of the spectrum_HThe three satisfy the relation: b ═ f_H-f_LEach sub-channel having a bandwidth of B₀2B/(k +1), time-bandwidth product factor c and pulse duration T_sBandwidth of each sub-channel B₀The three satisfy the relation: c ═ pi B₀T_sThe number of the elliptic spherical wave function pulses of each sub-channel is

② determining a fitting matrix D

According to the PSWF pulse time-bandwidth product factor c, the formulaDetermining the number of Legendre polynomials required for the PSWF pulse, whereinRepresenting rounding up, c is the time-bandwidth product of the PSWF, and e is the base of the natural logarithm;

establishing a characteristic matrix according to the formulas (9) to (12), and solving the characteristic vector of the matrix to obtain a coefficient matrix B ═ beta of the normalized Legendre polynomial¹，β²,…,β^M]Thus, a PSWF pulse signal approximated by a normalized Legendre polynomial can be obtained;

for the obtained PSWF pulse signal, calculating a cross-correlation matrix C of the pulse:

and the cross-correlation matrix is diagonalized, i.e. X^TAnd CX ═ Λ, and a transformation matrix X of the diagonalization power of the cross-correlation matrix C is obtained.

According to the formula D ═ X^TAnd B, calculating a fitting matrix D.

Obtaining polynomial coefficient vector D of orthogonal PSWF pulse from each row vector of fitting matrix D_i；

And fourthly, according to the normalized Legendre polynomial fitting form of the orthogonal PSWF pulse: p is a normalized Legendre polynomial, thus completing the orthogonal PSWF pulse design, as shown in FIG. 3.

(2) Bandpass PSWF orthogonalization design

For the bandpass PSWF orthogonalization design, the disclosed orthogonal PSWF pulse generation flow is shown in fig. 4, and the pulse generation can be performed as follows.

Dividing channels: dividing the frequency spectrum width occupied by a communication channel, namely orthogonal pulse into k sub-channels with the same bandwidth and overlapped with each other in a uniform overlapping mode, wherein k is a positive integer larger than 0, and the overlapping size rho of the frequency spectrum of the adjacent sub-channels can be represented by the percentage of the frequency spectrum bandwidth overlapped by the two adjacent sub-channels in the sub-channel bandwidth, and the value range is the percentage larger than 0 and smaller than 100%;

setting parameters: setting parameters of orthogonal pulse, the frequency spectrum width B occupied by orthogonal pulse, and the frequency spectrum lower limit frequency f_LAnd the upper frequency f of the spectrum_HDetermining the division number k of sub-channels, the time-bandwidth product factor c of an elliptic spherical wave function and the frequency spectrum overlapping degree rho, and determining the orthogonal pulse frequency spectrum width B and the frequency spectrum lower limit frequency f_LAnd the upper frequency f of the spectrum_HThe three satisfy the relation: b ═ f_H-f_LOrthogonal pulse spectral width B, sub-band bandwidth B₀The three degrees of overlapping with the frequency spectrum rho satisfy the relational expression: b ═ [ (1- ρ) k + ρ]B₀When the sub-channels are overlapped by 50%, the bandwidth of each sub-channel of the sub-channel is B₀2B/(k + 1); time-bandwidth product factor c, pulse duration T and sub-channel bandwidth B₀The three satisfy the relation: c ═ pi B₀T, the number of the elliptic spherical wave function pulses of each sub-channel is

Thirdly, constructing a characteristic function and an integral equation: the method is to construct a characteristic function and an integral equation for each sub-channel, wherein the constructed characteristic function is h_k(t)＝2f_k,H sinc(2f_k,Ht)-2f_k,L sinc(2f_k,Lt) wherein f_k,L、f_k,HRespectively the lower limit and the upper limit of the frequency of the kth sub-channel, and the constructed integral equation isWherein λ is_kCharacteristic function h for solving integral equation for numerical algorithm based on eigenvalue decomposition_k(t) eigenvalues of the constructed matrix,. psi_k(c, t) is lambda_kCorresponding characteristic function, T is the value range [ -T/2, T/2]A time variable of (d);

solving an equation: respectively solving integral equations constructed by each sub-channel through a Parr numerical algorithm based on characteristic value decomposition, and obtaining elliptic spherical wave function pulses of each sub-channel by taking characteristic functions corresponding to the first m maximum characteristic values;

orthogonalization based on cross-correlation matrix diagonalization: the method is characterized in that the cross-correlation matrix of each sub-channel PSWF pulse participating in orthogonalization design is calculated, and the cross-correlation matrix is subjected to diagonalization transformation, so that the orthogonalization of the PSWF pulse is realized.

Calculating a cross-correlation matrix C of the bandpass PSWF psi participating in the orthogonalization design:

the cross-correlation matrix is diagonalized, i.e. X^TAnd CX ═ Λ, and a transformation matrix X of the diagonalization power of the cross-correlation matrix C is obtained. Matrix multiplying the transposed form of the transformation matrix X with the pulse psi to obtain an orthogonal PSWF pulse psi', i.e. psi ═ X^Tψ. Thereby realizing the orthogonal design of the band-pass PSWF.

Claims

1. An orthogonal PSWF pulse design method based on cross-correlation matrix diagonalization is an orthogonal PSWF pulse design method, and is characterized in that a cross-correlation matrix C of PSWF pulses psi participating in orthogonal design is calculated, and the cross-correlation matrix C is used for calculating the cross-correlation matrix CWherein c is_i,jAs a function of the cross-correlation of the ith and jth pulses, i.e.The cross-correlation matrix is diagonalized, i.e. X^TCX ═ Λ, obtaining an orthogonal matrix X, wherein Λ is a diagonal matrix, and then passing through a transposed matrix X of the orthogonal matrix X^TMatrix multiplication with PSWF pulse ψ: x^Tψ to thereby realize orthogonalization of the PSWF pulse ψ, which may be either a baseband PSWF pulse or a band-pass PSWF.

2. The method of claim 1, wherein the designing of the quadrature PSWF pulse based on the diagonalization of the cross-correlation matrix is: a Legendre polynomial weighting coefficient matrix B of the PSWF pulse is obtained based on a numerical solving algorithm of normalized Legendre polynomial approximation, a cross-correlation matrix C of the PSWF pulse psi participating in orthogonalization design is calculated, and the cross-correlation matrix is subjected to diagonalization transformation: x^TObtaining an orthogonal matrix X by CX ═ Λ, wherein Λ is a diagonal matrix, and then transposing the orthogonal matrix X by the orthogonal matrix X^TMatrix multiplication is carried out on the pulse polynomial weighting coefficient matrix B of the PSWF pulse: x^TB, obtaining a weighted summation matrix D ═ X of the PSWF pulse^TB, thereby obtaining a normalized Legendre polynomial fit of the orthogonal PSWF pulse: and psi' ═ DP, namely, an orthogonalized design of the PSWF pulse psi is realized, where P is a normalized Legendre polynomial.

3. The cross-correlation matrix diagonalization-based orthogonal PSWF pulse design method of claim 1, wherein the bandpass PSWF pulse orthogonalization design comprises the steps of: dividing channels, setting parameters, constructing a characteristic function and an integral equation, solving the equation, and carrying out diagonalization based on a cross-correlation matrix;

the channel division is to divide the frequency spectrum width occupied by the communication channel, namely the orthogonal pulse, into a plurality of sub-channels which have the same bandwidth and are mutually overlapped in a uniform overlapping mode;

the parameter setting is to set the parameters of the orthogonal pulse, the frequency spectrum width B occupied by the orthogonal pulse, and the lower limit frequency f of the frequency spectrum_LAnd the upper frequency f of the spectrum_HSub-channel division number k, ellipseDetermining a time bandwidth product factor c and a frequency spectrum overlapping degree rho of the spherical wave function;

the characteristic function and the integral equation are constructed by constructing an integral equation definition formula of the band-pass PSWF for each sub-channel;

solving the equation refers to respectively solving the approximate numerical solution of the band-pass PSWF of each sub-channel by adopting a Parr numerical algorithm;

orthogonalization based on diagonalization of the cross-correlation matrix is: by calculating the cross-correlation matrix C of each sub-channel PSWF pulse psi participating in the orthogonalization design, the cross-correlation matrix is subjected to diagonalization transformation, namely X^TCX ═ Λ, obtaining an orthogonal matrix X, wherein Λ is a diagonal matrix, and then passing through a transposed matrix X of the orthogonal matrix X^TMatrix multiplication with PSWF pulse ψ: x^TPsi, thereby realizing the orthogonalization design of the PSWF pulse psi.